Can different paths in spacetime have the same separation?

[Anamitra]

You don't have to go below the event horizon. Please stay above it and consider changes in the coefficients:
{{(}{1}{-}{2M}{/}{r}{)}} and {{(}{1}{-}{2M}{/}{r}{)}}^{-1}


Such changes are quite possible if some high density erratic mass distribution comes near the mass M. The resultant metric will be quite different[the coefficients will undergo a huge change in their functional expression].A space like path may get converted into a timelike one and vice versa. [It is not necessary to make dt=0 for such considerations]

[The time dependence of a gravitational field is related to the changes in the values/expressions of the metric coefficients]

 

[Anamitra]

Regarding Time dependence:I can always choose a frame of reference where the spatial coordinate labels [x,y,z] do not change with time for my inferences.The values/expressions for the metrics do change due to gravitational effects.PAllen and others can always choose a frame where the coordinates are changing. The physical nature of the conclusions should not change.

A simple illustration:
I am in a laboratory at A(x1,y1,z1) . Pallen and yuiop are at different one B(x2,y2,z2). A few minutes ago we had spacelike paths between the two labs. Now we have timelike paths thaks to the gravitational effects.

Would the physical nature of the conclusions change if by some suitable transformation the spatial coordinates are made to vary with time?

 

[PAllen]

Let us first come back to the basic issue---whether there there can be an intercoversion between a spacelike path and a timelike/null path by the effect of gravity[Due to changes in the values/expressions representing the metric coefficients in a time dependent field]
{ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{2}

This is a strange way to word it. A given spacetime path is, by definition, fixed in time and character. I think what you must mean is that a similar (e.g same coordinate slope) coordinate path in different regions of spacetime can be spacelike in one region and timelike in another. That is obviously true, and whenever it is true, a trivial coordinate transform can convert it to the simpler case of a coordinate axis having different character in different regions of spacetime.

The value of the metric coefficients can always change by the action of gravity---this is a known fact.The sign of ds^2 can also change[I am referring to such a possibility here] . By choosing a different coordinate system you can never change the physical consequences.

Correct. Change in coordinate system will never change an invariant like the spacelike/timelike character of a path.

Even if you consider some weird geometry with unusual signs for the metric coefficients[in case such an action is possible]or even if you apply any other type of contrivance, the huge number of cases[in relation to the interconversion] indicated in the last paragraph cannot be ruled out.

Of course. No one disagreed with this. From my point of view, I was simply proposing the simplest example of this.

The light cone mechanism has been clearly depicted in the following link:
https://www.physicsforums.com/showpost.php?p=3061481&postcount=38

Many other side issues have come up in this thread.I will definitely address them[and I have been addressing them]

Actually, they are mostly all the same issue. You think they are different for superficial reasons (like whether dt=0 or not).

 

[Dale]

Let us first come back to the basic issue---whether there there can be an intercoversion between a spacelike path and a timelike/null path by the effect of gravity

The answer to that is very clearly and definitively, "no".

What can happen is that there may be two different paths which differ only by a constant offset of one of the coordinates (e.g. t) and one of these two different paths may be spacelike while the other is timelike.

 

[Dale]

A simple illustration:
I am in a laboratory at A(x1,y1,z1) . Pallen and yuiop are at different one B(x2,y2,z2). A few minutes ago we had spacelike paths between the two labs. Now we have timelike paths thaks to the gravitational effects.

If you let y1=y2=0 and z1=z2=0 then this is exactly the example I gave previously. Note that the spacelike and timelike paths are different paths because the t coordinate differs by a few minutes.

 

[PAllen]

Regarding Time dependence:I can always choose a frame of reference where the spatial coordinate labels [x,y,z] do not change with time for my inferences.The values/expressions for the metrics do change due to gravitational effects.PAllen and others can always choose a frame where the coordinates are changing. The physical nature of the conclusions should not change.

A simple illustration:
I am in a laboratory at A(x1,y1,z1) . Pallen and yuiop are at different one B(x2,y2,z2). A few minutes ago we had spacelike paths between the two labs. Now we have timelike paths thaks to the gravitational effects.

Would the physical nature of the conclusions change if by some suitable transformation the spatial coordinates are made to vary with time?

This is getting at your confusion. You can speak of a space *time* path being spacelike. What you mean is that the path between:

(t1,x1,y1,z1) and (t1,x2,y2,z2) maintaining t=t1 is spacelike,

while the path:

(t2,x1,y1,z1) and (t2,x2,y2,z2) maintaining t=t2 is timelike.

These are two completely independent paths through spacetime, and the situation implies, mostly, that the meaning of the coordinates *has* changed. There is, presumably, a family of spacelike paths connecting the world lines of the two labs at different proper time points along the world lines. The situation above simply means the the coordinate representation of these paths looks very different at different points along the worldlines.

 

[Anamitra]

Let us first come back to the basic issue---whether there there can be an intercoversion between a spacelike path and a timelike/null path by the effect of gravity[Due to changes in the values/expressions representing the metric coefficients in a time dependent field]
{ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{2}

This is a strange way to word it. A given spacetime path is, by definition, fixed in time and character. I think what you must mean is that a similar (e.g same coordinate slope) coordinate path in different regions of spacetime can be spacelike in one region and timelike in another. That is obviously true, and whenever it is true, a trivial coordinate transform can convert it to the simpler case of a coordinate axis having different character in different regions of spacetime.

A spacetime path remains spacetime in time independent gravitational fields. Their nature remains invariant wrt coordinate transformation[in all types of fields time dependent or independent].In my illustration the coordinate system is not being changed or transformed.We are considering changes in the metric coefficients in the same coordinate system(t,x,y,z). Conclusions should remain unchanged for all other frames once the change [in tne metric coefficients]has taken place.

Correct. Change in coordinate system will never change an invariant like the spacelike/timelike character of a path.

This does not come in the way of my arguments

 

[PAllen]

Let's try a particular example in Schwarzschild metric. Assume dx2 and dx3 are zero so we are considering the two dimensional radial case so:

ds^2 = (1-2M/r)dt^2 - (1-2M/r)^{-1} dr^2

For a timelike path, ds^2 is positive. Below the event horizon (say r=M) a path with dt=0 is a valid timelike path because:

ds^2 = - (1-2M/M)^{-1} dr^2 = +dr^2

Can we calculate a permissible velocity, momentum and energy of such a particle?

Of course. You need to specifiy a path. If you say dt=0 and r=M, you have an event not a path. You can say, e.g. r varies from M to .9M, while t=t0. At r=M, you get a 4 velocity of (vt,vr)=(0,1), a contravariant momentum of (0,m); and if you take the norm of the momentum using the metric, you get m as rest energy (of course).

 

[Anamitra]

This is getting at your confusion. You can speak of a space *time* path being spacelike. What you mean si that the path between:

(t1,x1,y1,z1) and (t1,x2,y2,z2) maintaining t=t1 is spacelike,

while the path:

(t2,x1,y1,z1) and (t2,x2,y2,z2) maintaining t=t2 is timelike.

These are two completely independent paths through spacetime, and the situation implies, mostly, that the meaning of the coordinates *has* changed. There is, presumably, a family of spacelike paths connecting the world lines of the two labs at different proper time points along the world lines. The situation above simply means the the coordinate representation of these paths looks very different at different points along the worldlines.

This part is absolutely OK. Thanks for that PAllen!
Now we are thinking of two spacetime paths A and B where B is a subset of A.t for initial point of A is greater than t for initial point of B. The part between initial point of A and initial point of B is time like and the rest is spacelike when the observer is at A.When the observer reaches the initial point of B he is amazed to find that the rest of the journey can be carried out since the remailing path has become timelike due to gravity!

The example in the following link is in tune with what you are saying!
https://www.physicsforums.com/showpost.php?p=3061386&postcount=36

 

[PAllen]

This part is absolutely OK. Thanks for that PAllen!
Now we are thinking of two spacetime paths A and B where B is a subset of A.t for initial point of A is greater than t for initial point of B. The part between initial point of A and initial point of B is time like and the rest is spacelike when the observer is at A.When the observer reaches the initial point of B he is amazed to find that the rest of the journey can be carried out since the path has become timelike!

The example in the following link is in tune with what you are saying!
https://www.physicsforums.com/showpost.php?p=3061386&postcount=36

Sorry, but this doesn't really make sense. The path from beginning of A to B cannot be the the path of any observer through spacetime. This path is truly like the following (and can be converted to it by coordinate transform):

imagine a born rigid ruler at some t=t0; this ruler represents the beginning of A to beginning of B; Now imagine the worldline of some observer that intersects the ruler at the beginning of B. This is the rest of this mixed spacetime path. You are simply drawing a path through spacetime that follows a ruler, then follows a worldline. This is not a very meaningful path, but you can define it.

[Edit: I accidentally reversed which part of the path is which from what you proposed, but the idea is the same see my later reply for additional point]

 

[Anamitra]

PAllen has clearly misread/misinterpreted the thought experiment described. Let me help him in getting the matter clarified.

WE consider a spacetime curve running from X[t1,x1,y1,z1] to Z[t3,x3,y3,z3] via Y[t2,x2,y2,z2]. The curve between X and Y is time like and the curve between Y and Z is spacelike . t1<t2

In the time t1 to t2 the observer reaches from X to Y along the timelike curve with the expectation that he will find a space like curve between Y and Z.[or may be observers at the spatial position of Y standing for a long time before his advent will inform him about the nature of the path ahead of him--and how it has changed]. The apparently unreachable spacetime point is now reachable!

 

[PAllen]

This part is absolutely OK. Thanks for that PAllen!
Now we are thinking of two spacetime paths A and B where B is a subset of A.t for initial point of A is greater than t for initial point of B. The part between initial point of A and initial point of B is time like and the rest is spacelike when the observer is at A.When the observer reaches the initial point of B he is amazed to find that the rest of the journey can be carried out since the remailing path has become timelike due to gravity!

The example in the following link is in tune with what you are saying!
https://www.physicsforums.com/showpost.php?p=3061386&postcount=36

I hope I can get closer to your misunderstanding. If B is a spacetime path it simply is spacelike. You cannot talk about a spacetime path changing nature. You can talk about *different* spacetime paths that have similar coordinate representations having different character (spacelike/timelike). The clause:

"and the rest is spacelike when the observer is at A"

is utterly meaningless. The rest is or isn't spacelike, there is no 'when' about it.

The spacetime path you describe simply joins an observer going from beginning of A to beginning of B and encountering the end of a ruler at B.

 

[PAllen]

PAllen has clearly misread/misinterpreted the thought experiment described. Let me help him in getting the matter clarified.

WE consider a spacetime curve running from X[t1,x1,y1,z1] to Z[t3,x3,y3,z3] via Y[t2,x2,y2,z2]. The curve between X and Y is time like and the curve between Y and Z is spacelike . t1<t2

In the time t1 to t2 the observer reaches from X to Y along the timelike curve with the expectation that he will find a space like curve between Y and Z.[or may be observers at the spatial position of Y standing for a long time before his advent will inform him about the nature of the path ahead of him--and how it has changed]. The apparently unreachable spacetime point is now reachable!

This is impossible. The nature of Y to Z cannot 'change', this isn't remotely meaningful. What the path X-Y-Z represents is that an observer following X to Y encounters a ruler at Y, going from Y to Z.

 

[Anamitra]

You can always take it in this way:Observers have been standing at the spatial position of y from time<t2.They knew very well that the spacetime point Z had a spacelike separation for time<t2.This information was transmitted to the observer at the spacetime point X at some suitable time<t2[or may be when the observer [initialy at X ] is on the way to Y!

The observer ,when he arrives at Y is amazed to find that the path ahead of him has become timelike!
Observers standing at the same spatial point have the same notion at time=t2
[Spacetime point Y is the same for all observers instantaneously,when the moving observer arrives there]

 

[PAllen]

You can always take it in this way:Observers have been standing at the spatial position of y from time<t2.They knew very well that the spacetime point Z had a spacelike separation for time<t2.This information was transmitted to the observer at the spacetime point X at some suitable time<t2[or may be when the observer [initialy at X ] is on the way to Y!

The observer ,when he arrives at Y is amazed to find that the path ahead of him has become timelike!
Observers standing at the same spatial point have the same notion at time=t2
[Spacetime point Y is the same for all observers instantaneously,when the moving observer arrives there]

What you can really say is that an observer at some position notes there are always time like paths he can initiate (firing bullets say), constant time spacelike paths (there's a ruler sitting next to him), and time varying spacelike paths (someone whisking a flashlight back and forth from a nearby building, fast enough so the light spot on the ground is moving faster than c). This is the physics of what's going on. Then, for some chosen coordinate system (and this is purely a feature of the chosen coordinate system), it happens that after a while, the coordinate description that had applied to the flashlight paths now applies to the bullet paths. Nothing physical has changed. Using different coordinates, no such 'anomaly' would be seen.

 

[JesseM]

You can always take it in this way:Observers have been standing at the spatial position of y from time<t2.They knew very well that the spacetime point Z had a spacelike separation for time<t2.

By "point Z" do you mean a point in space (which persists over time) or a point in spacetime (an instantaneously brief localized event)? When physicists talk about a "spacelike separation" between points, they're talking about points in spacetime. If this is what you're doing, what two events are you saying had a spacelike separation? Also, note that although in SR one can talk about the separation between two points (with it understood that we're looking at a straight line in spacetime between them), in GR one really needs to specify a path through spacetime to say if it's spacelike or timelike or lightlike, since there are multiple paths between any given pair of events and none of them uniquely represent "the" separation between them.

This information was transmitted to the observer at the spacetime point X at some suitable time<t2[or may be when the observer [initialy at X ] is on the way to Y!

The observer ,when he arrives at Y is amazed to find that the path ahead of him has become timelike!

What do you mean "the path ahead of him"? "Ahead" in what sense? Ahead in some spatial direction, or ahead in his future light cone? Your scenario is really difficult to understand in words and I doubt anyone else is understanding it much better than I am, it would help if you either drew a spacetime diagram or gave a numerical example, or at least described all the worldlines and events more carefully, being sure to distinguish between points and paths in space and points and paths in spacetime, and perhaps also specifying which points in spacetime like in the future light cones of other points and which pairs of points are not in each other's past or future light cones (so there is no timelike path through spacetime between them, and no signal traveling at the speed of light or slower could get from one point to the other).

Observers standing at the same spatial point have the same notion at time=t2
[Spacetime point Y is the same for all observers instantaneously,when the moving observer arrives there]

You said before y was a spatial position, now you're saying it's a spacetime point? They are two very different concepts? Again, with a spatial position you can talk about the same position at different times, but a spacetime point is an instantaneously brief event.

 

[Anamitra]

I believe that I have been misread once more. So I am trying to clarify my stand:

WE consider a spacetime curve running from X[t1,x1,y1,z1] to Z[t3,x3,y3,z3] via Y[t2,x2,y2,z2]. The curve between X and Y is time like and the curve between Y and Z is EXPECTEDLY spacelike . t1<t2

In the time t1 to t2 the observer reaches from X to Y along the timelike curve with the expectation that he will find a space like curve between Y and Z.[or may be observers at the spatial position of Y standing for a long time before his advent will inform him about the nature of the path ahead of him--and how it has changed]. The apparently unreachable spacetime point is now reachable!

The term EXPECTEDLY has been explained below:

You can always take it in this way:Observers have been standing at the spatial position of y from time<t2.They knew very well that the spacetime point Z had a spacelike separation for time<t2.They could well expect the path remain spacelike for t=t2. That is the observes are expecting a spacelike separation between Y and Z which are spacetime points.This information was transmitted to the observer at the spacetime point X at some suitable time<t2[or may be when the observer [initially at X ] is on the way to Y!

The observer ,when he arrives at Y is amazed to find that the path ahead of him has become timelike!
Observers standing at the same spatial point have the same notion at time=t2
[Spacetime point Y is the same for all observers instantaneously,when the moving observer arrives there]

 

[Anamitra]

One may,in many circumstances envisage[they may pre-calculate or they may be pre-informed] the nature of separation between a pair of spacetime points[events] before the occurrence of the events, from the present situation of the metric coefficients--their expressions/values.But when the events occur they may find that the nature of separation has changed---spacelike paths have time like or vice-versa.

 

[JesseM]

The term EXPECTEDLY has been explained below:

You can always take it in this way:Observers have been standing at the spatial position of y from time<t2.They knew very well that the spacetime point Z had a spacelike separation for time<t2.

A spacelike separation from what?

They could well expect the path remain spacelike for t=t2.

What is this "path" you are talking about? What set of events does it pass through? You say "remain spacelike", does that mean some section of this path is spacelike, and if so what section is that? You never define any of your terms clearly!

That is the observes are expecting a spacelike separation between Y and Z which are spacetime points.

Why do they "expect" that? Are they ignorant of the metric?

This information was transmitted to the observer at the spacetime point X at some suitable time<t2[or may be when the observer [initially at X ] is on the way to Y!

What information?

 

[PAllen]

One may,in many circumstances envisage the nature of separation between a pair of spacetime points[events] before the occurrence of the events, from the present situation of the metric coefficients--their expressions/values.But when the events occur they may find that the nature of separation has changed---spacelike paths have time like or vice-versa.

This really makes no sense. You are looking at time varying metric components expressed in some specific coordinates, and trying to explain it this way. This is not a correct explanation. A correct explanation of how this situation would be perceived is given in my post #105.

 

[Anamitra]

Please go through the posts #107 and #108, Jesse

 

[JesseM]

One may,in many circumstances envisage the nature of separation between a pair of spacetime points[events] before the occurrence of the events, from the present situation of the metric coefficients--their expressions/values.But when the events occur they may find that the nature of separation has changed---spacelike paths have time like or vice-versa.

In GR you can predict what the metric coefficients will be in the future if you know the coefficients along with the distribution of matter/energy in the present. And in any case I don't think you can meaningfully define a coordinate system on a region of spacetime where you don't know the metric, and without a coordinate system how can you even pinpoint specific events and paths in this region in order to talk about whether the paths to the events are spacelike or timelike?

 

[JesseM]

Please go through the posts #107 and #108, Jesse

My post #109 was directly in response to #107, I asked questions because I found your description there completely unclear (and I doubt anyone else reading this thread could follow it either). Post #108 is unclear as well, see my response in #112. If you want to be understood, it might help if you would try your best to give specific answers to the questions I ask.

 

[Anamitra]

In GR you can predict what the metric coefficients will be in the future if you know the coefficients along with the distribution of matter/energy in the present. And in any case I don't think you can meaningfully define a coordinate system on a region of spacetime where you don't know the metric, and without a coordinate system how can you even pinpoint specific events and paths in this region in order to talk about whether the paths to the events are spacelike or timelike?

The important aspect to consider is the finite speed of signal transmission!
I can always get informed about the nature of metrics at different points.Information about the changed state /changed values of the metrics will reach me later. There is a period of ignorance which may be a hundred years or more

[At each spatial point we may preserve information artificially or naturally for a future assessment----one may assume that]

 

[JesseM]

The important aspect to consider is the finite speed of signal transmission!
I can always get informed about the nature of metrics at different points.Information about the changed state /changed values of the metrics will reach me later. There is a period of ignorance which may be a hundred years or more

Again, you can predict what the metric will be like even in regions you can't get an actual signal from. If you don't know enough about a region to predict the metric, then it seems to me the only "events" you can talk about in this region are fairly generic ones like "the future decay of some particle I saw earlier" or "the event of some clock showing a time T later than the time it showed the most recent moment I saw it". How would you even describe a specific path to speculate about whether it'll be spacelike or timelike in that region?

I suppose if you had a family of clocks connected by flexible springs filling space, then you could define a coordinate system where each clock had a constant position coordinate and its reading defined a time coordinate, so then you could talk about coordinates even in regions where you didn't know the metric, and define a "path" in terms of those coordinates...but if you didn't know the metric it's hard to see how you could have much basis for "expecting" that a particular path would be spacelike or timelike (except a special cases like a path of constant position coordinate, which would just be the worldline of some clock and must therefore be timelike)

 

[Anamitra]

We are considering a set of time slices corresponding to t1 ,t2,t3...t(n-1),tn [The time coordinates are in increasing order and time remains constant for each slice]
When at (t1,xA,yA,zA), I may try to visualize/pre-assess the nature of separation between the events (t(n-1),xA,yA,zA) and (tn,xB,yB,zB) along some coordinate curve[the coordinate labels are not changing]. The curve runs between the two events on two specified time slices. I know that the metrics will be changing. AS I move from (t1,xA,yA,zA) towards (t(n-1),xA,yA,zA) there may be a huge number of instants for which I may not be able to predict the nature of the curve between (t(n-1),xA,yA,zA) and (tn,xB,yB,zB)--they may turn out to be of any type timelike ,spacelike or null.

The curve has the same set of coordinate points for all predictions

We may also think in terms of parallel ensembles[I mean to say groups of time slices of the type mentioned in the first paragraph]with the surfaces corresponding to t1 as identical.We have different sets of time slices for the different sets of metric coefficients[coordinate labels are the same for the curve].And different situations are realizable -----spacelike,null or timelike connections between the same points and along the same coordinate curves .[for different ensembles]

 

[Anamitra]

I may not be able to predict the exact shape/nature of future time slices and hence the nature of separation between points lying on them[separate time slices] could be anything!

[I may have to wait a very long time for any correct prediction]

 

[Dale]

Anamitra, this is completely silly. The fact that we may be surprised about something due to ignorance on the subject is not something new to GR.

If you are ignorant of the metric in some region of spacetime then you are also ignorant of whether or not a given path through that region is timelike or spacelike. Again, a path does not change from timelike to spacelike due to the influence of gravity. If the metric is unknown then so is the nature of the path. In fact, if the metric is unknown then whether or not a given set of coordinates is even valid is also unknown.

 

[Anamitra]

Anamitra, this is completely silly. The fact that we may be surprised about something due to ignorance on the subject is not something new to GR.

If you are ignorant of the metric in some region of spacetime then you are also ignorant of whether or not a given path through that region is timelike or spacelike. Again, a path does not change from timelike to spacelike due to the influence of gravity. If the metric is unknown then so is the nature of the path. In fact, if the metric is unknown then whether or not a given set of coordinates is even valid is also unknown.

You may just think of the situation[time dependent fields depicted in #116 and #117] in contrast against stationary fields.

 

[PAllen]

We are considering a set of time slices corresponding to t1 ,t2,t3...t(n-1),tn [The time coordinates are in increasing order and time remains constant for each slice]
When at (t1,xA,yA,zA), I may try to visualize/pre-assess the nature of separation between the events (t(n-1),xA,yA,zA) and (tn,xB,yB,zB) along some coordinate curve[the coordinate labels are not changing]. The curve runs between the two events on two specified time slices. I know that the metrics will be changing. AS I move from (t1,xA,yA,zA) towards (t(n-1),xA,yA,zA) there may be a huge number of instants for which I may not be able to predict the nature of the curve between (t(n-1),xA,yA,zA) and (tn,xB,yB,zB)--they may turn out to be of any type timelike ,spacelike or null.

The curve has the same set of coordinate points for all predictions

We may also think in terms of parallel ensembles[I mean to say groups of time slices of the type mentioned in the first paragraph]with the surfaces corresponding to t1 as identical.We have different sets of time slices for the different sets of metric coefficients[coordinate labels are the same for the curve].And different situations are realizable -----spacelike,null or timelike connections between the same points and along the same coordinate curves .[for different ensembles]

Yes, you could do something like this. Note:

1) You seem to be attaching great physical significance to the coordinate labels. This is not meaningful.

2) Because of (1), you fail to see that what you describe is just because of the expression of the metric in this particular coordinate system. Please try to understand my post #105. That is how this situation would be physically experienced.

3) Take particular note that if you change coordinates to one more natural for this particular observer's world line, you wouln't observe any unusual change in coordinate properties.